自然法則の微分方程式微分方程式の解微分方程式の解、指数関数、三角関数、正弦と余弦

微分方程式演習
〈理工系の数学入門コース/演習新装版〉
楽天ブックス
Yahoo!

学習環境

微分方程式演習〈理工系の数学入門コース/演習新装版〉 (和達三樹(著)、矢嶋徹(著)、岩波書店)の第1章(自然法則の微分方程式)、1-3(微分方程式の解)、問題2の解答を求めてみる。

\frac{d N}{dt} = \frac{N_{\infty}}{{(1 + C_{1} e^{- {µ N}_{\infty} t})}^{2}} C_{1} µ N_{\infty} e^{- {µ N}_{\infty} t} = µ N C_{1} µ e^{- {µ N}_{\infty} t}

\begin{array}{l} N + N C_{1} e^{- {µ N}_{\infty} t} = N_{\infty} \\ N C_{1} e^{- {µ N}_{\infty} t} = N_{\infty} - N \end{array}

\frac{d N}{dt} = µ N (N_{\infty} - N)

α > Ω_{g}

の場合、

\frac{d θ}{dt} = C_{1} (- α + β) e^{(- α + β) t} + C_{2} (- α - β) e^{(- α - β) t} + \frac{Ω_{e}^{2}}{2 α} \sin Ω_{g} t

\frac{d^{2} θ}{{dt}^{2}} = C_{1} {(- α + β)}^{2} e^{(- α + β) t} + C_{2} {(- α - β)}^{2} e^{(- α - β) t} + \frac{Ω_{e}^{2} Ω_{g}}{2 α} \cos Ω_{g} t

\frac{d^{2} θ}{{dt}^{2}} + 2 α \frac{d θ}{dt} + Ω_{g}^{2} θ

\begin{array}{l} = C_{1} ({(- α + β)}^{2} + 2 α (- α + β) + Ω_{g}^{2}) e^{(- α + β) t} \\ + C_{2} ({(- α - β)}^{2} + 2 α (- α - β) + Ω_{g}^{2}) e^{(- α - β) t} \\ + \frac{Ω_{e}^{2} Ω_{g}}{2 α} \cos Ω_{g} t + Ω_{e}^{2} \sin Ω_{g} t - \frac{Ω_{e}^{2} Ω_{g}}{2 α} \cos Ω_{g} t \end{array}

= C_{1} (- α^{2} + β^{2} + Ω_{g}^{2}) e^{(- α + β) t} + C_{2} (- α^{2} + β^{2} + Ω_{g}^{2}) e^{(- α - β) t} + Ω_{e}^{2} \sin Ω_{g} t

= (- α^{2} + β^{2} + Ω_{g}^{2}) (C_{1} e^{(- α + β) t} + C_{2} e^{(- α - β) t}) + Ω_{e}^{2} \sin Ω_{g} t

- α^{2} + β^{2} + Ω_{g}^{2} = - α^{2} + | α^{2} - Ω_{g}^{2} | + Ω_{g}^{2} = - α^{2} + α^{2} - Ω_{g}^{2} + Ω_{g}^{2} = 0

よって、

\frac{d^{2} θ}{d t^{2}} + 2 α \frac{d θ}{dt} + Ω_{g}^{2} θ = Ω_{e}^{2} \sin Ω_{g} t

α = Ω_{g}

の場合、

\begin{array}{l} \frac{d θ}{dt} = - C_{1} α e^{- α t} + C_{2} (e^{- α t} - t α e^{- α t}) + \frac{Ω_{e}^{2}}{2 α} \sin α t \\ = - C_{1} α e^{- α t} + C_{2} (1 - t α) e^{- α t} + \frac{Ω_{e}^{2}}{2 α} \sin α t \end{array}

\begin{array}{l} \frac{d^{2} θ}{{dt}^{2}} = C_{1} α^{2} e^{- α t} + C_{2} (- α e^{- α t} - α e^{- α t} + t α^{2} e^{- α t}) + \frac{Ω_{e}^{2}}{2} \cos α t \\ = C_{1} α^{2} e^{- α t} + C_{2} (- 2 α + t α^{2}) e^{- α t} + \frac{Ω_{e}^{2}}{2} \cos α t \end{array}

\frac{d^{2} θ}{{dt}^{2}} + 2 α \frac{d θ}{dt} + Ω_{g}^{2} θ

= \frac{d^{2} θ}{{dt}^{2}} + 2 α \frac{d θ}{dt} + α^{2} (C_{1} e^{- α t} + C_{2} t e^{- α t} - \frac{Ω_{e}^{2}}{2 α^{2}} \cos α t)

\begin{array}{l} = C_{1} e^{- α t} (α^{2} - 2 α^{2} + α^{2}) + C_{2} e^{- α t} (- 2 α + t α^{2} + 2 α - 2 t α^{2} + α^{2} t) \\ + \frac{Ω_{e}^{2}}{2} \cos α t + Ω_{e}^{2} \sin α t - \frac{Ω_{e}^{2}}{2} \cos α t \end{array}

= Ω_{e}^{2} \sin Ω_{g} t

α < Ω_{g}

の場合、

\begin{array}{l} \frac{d θ}{dt} = C_{1} (- α e^{- α t} \cos β t - e^{- α t} β \sin β t) \\ + C_{2} (- α e^{- α t} \sin β t + e^{- α t} β \cos β t) \\ + \frac{Ω_{e}^{2}}{2 α} \sin Ω_{g} t \\ = C_{1} e^{- α t} (- α \cos β t - β \sin β t) \\ + C_{2} e^{- α t} (- α \sin β t + β \cos β t) \\ + \frac{Ω_{e}^{2}}{2 α} \sin Ω_{g} t \end{array}

\begin{array}{l} \frac{d^{2} θ}{{dt}^{2}} = C_{1} (- α (- α e^{- α t} \cos β t - e^{- α t} β \sin β t) - β (- α e^{- α t} \sin β t + e^{- α t} β \cos β t)) \\ + C_{2} (- α (- α e^{- α t} \sin β t + e^{- α t} β \cos β t) + β (- α e^{- α t} \cos β t - e^{- α t} β \sin β t)) \\ + \frac{Ω_{e}^{2} Ω_{g}}{2 α} \cos Ω_{g} t \\ = C_{1} e^{- α t} (α^{2} \cos β t + α β \sin β t + α β \sin β t - β^{2} \cos β t) \\ + C_{2} e^{- α t} (α^{2} \sin β t - α β \cos β t - α β \cos β t - β^{2} \sin β t) \\ + \frac{Ω_{e}^{2} Ω_{g}}{2 α} \cos Ω_{g} t \end{array}

\frac{d^{2} θ}{{dt}^{2}} + 2 α \frac{d θ}{dt} + Ω_{g}^{2} θ

\begin{array}{l} = C_{1} e^{- α t} (- α^{2} \cos β t - β^{2} \cos β t + Ω_{g}^{2} \cos β t) \\ + C_{2} e^{- α t} (- α^{2} \sin β t - β^{2} \sin β t + Ω_{g}^{2} \sin β t) \\ + \frac{Ω_{e}^{2} Ω_{g}}{2 α} \cos Ω_{g} t + Ω_{e}^{2} \sin Ω_{g} t - \frac{Ω_{e}^{2} Ω_{g}}{2 α} \cos Ω_{g} t \end{array}

\begin{array}{l} = C_{1} e^{- α t} (- α^{2} \cos β τ - (Ω_{g}^{2} - α^{2}) \cos β t + Ω_{g}^{2} \cos β t) \\ + C_{2} e^{- α t} (- α^{2} \sin β t - (Ω^{2} - α^{2}) \sin β t + Ω_{g}^{2} \sin β t) \\ + Ω_{e}^{2} \sin Ω_{g} t \end{array}

= Ω_{e}^{2} \sin Ω_{g} t

\frac{d v}{dt} + Γ v (t)

= - C_{1} Γ e^{- Γ t} - Γ e^{- Γ t} \int_{t_{0}}^{t} f (t') e^{Γ t'} dt' + e^{- Γ t} f (t) e^{Γ t} + Γ v (t)

= - C_{1} Γ e^{- Γ t} - Γ e^{- Γ t} \int_{t_{0}}^{t} f (t') e^{Γ t} dt + f (t) + Γ v (t)

= - Γ v (t) + f (t) + Γ v (t)

= f (t)

\frac{d^{2}}{d t^{2}} x (t)

= \frac{d}{dt} (- C_{1} Ω \sin Ω t + C_{2} Ω \cos Ω t)

= - C_{1} Ω^{2} \cos Ω t - C_{2} Ω^{2} \sin Ω t

\frac{d^{2}}{{dt}^{2}} \int_{t_{0}}^{t} f (t') \sin Ω (t' - t) dt'

= - Ω \frac{d}{dt} \int_{t_{0}}^{t} f (t') \cos Ω (t' - t) dt'

= - Ω (f (t) + Ω \int_{t_{0}}^{t} f (t') \sin Ω (t' - t) dt')

\frac{d^{2}}{{dt}^{2}} x (t) + Ω^{2} x (t)

\begin{array}{l} = - Ω^{2} (C_{1} \cos Ω t + C_{2} \sin Ω t) + f (t) + Ω \int_{t_{0}}^{t} f (t') \sin Ω (t' - t) dt' \\ + Ω^{2} (C_{1} \cos Ω t + C_{2} \sin Ω t) - Ω \int_{t_{0}}^{t} f (t') \sin Ω (t' - t) {dt}^{1} \end{array}

= f (t)

コード(Wolfram Language, Jupyter)

n[t_] := n0/(1+c1 Exp[-u n0 t])

n'[t] == u n[t] (n0 - n[t])

Simplify[%]

b = Sqrt[Abs[a^2-og^2]]

theta[t_] := If[
    a > og, c1 Exp[-a + b]t + c2 Exp[-a -b] t - oe^2/(2a og) Cos[og t],
    If[a == og,
       c1 Exp[-og t] + c2 t Exp[-og t] - oe^2 / (2 og^2) Cos[og t],
       c1 Exp[-a t] Cos[b t] + c2 Exp[-a t] Sin[b t] - oe^2 / (2 a og) Cos[og t]
    ]
]

theta''[t] + 2 a theta'[t] + og^2 theta[t] == oe^2 Sin[og t]

Simplify[%]

v[t_] := c1 Exp[-g t] + Exp[-g t] Integrate[f[t1] Exp[g t1], {t1, t0, t}]

v'[t] + g v[t] == f[t]

Simplify[%]

x[t_] := c1 Cos[o t] + c2 Sin[o t] - 1 / o Integrate[f[s] Sin[o (s - t)], {s, t0, t}]

x'[t]

x''[t]

% + o^2 x[t]

Simplify[%]